Animated Solution for Mathematics - Functions: Let $f:\mathbb{N}\to \mathbb{N}$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m,n\in \mathbb{N}$. If $f(6)=18$, then $f(2)\cdot f(3)$ is equal to :

Welcome, fellow traveler on the path to JEE mastery. Today, we are going to peel back the layers of a classic functional equation.

Functional equations are the riddles of the mathematical world—they don't give you the function explicitly; instead, they give you a rule, a behavior, or a 'personality' that the function must obey. Our function $f:\mathbb{N}\to \mathbb{N}$ is governed by the rule:

This is not just any rule; this is the Cauchy Functional Equation, the bedrock of linear behavior in mathematics.

Decoding the DNA of the Function

Imagine you are standing at the base of a staircase. The rule $f(m+n)=f(m)+f(n)$ tells us that the total height of a combined jump is simply the sum of the individual jumps.

If you jump $m$ steps and then $n$ steps, the total 'value' of that jump is the same as if you had jumped $m$ steps and $n$ steps separately. This is the definition of additivity.

It implies that the function is 'blind' to the order or grouping of the inputs; it only cares about the total magnitude. This is the hallmark of a linear relationship, and we suspect that $f(n)$ is simply a multiple of $n$.

Building the Ladder

We start with the smallest possible step. Let $m=1$ and $n=1$. Substituting these into our rule, we get:

Now, let's climb higher. To find $f(3)$, we can set $m=2$ and $n=1$. The rule gives us:

Since we already know $f(2)=2f(1)$, we substitute that in:

Do you see the rhythm? It is like a heartbeat. If we continued this logic, $f(4)$ would be $4f(1)$, and by the power of mathematical induction, for any natural number $n$, we can confidently state:

We have successfully reduced the entire function to a single unknown constant: $f(1)$.

Unlocking the Secret

Now, we turn to the specific clue provided by the problem: $f(6)=18$. Our general pattern tells us that:

Therefore, we have the equation $6\cdot f(1)=18$. This is the moment where the fog clears.

By dividing both sides by $6$, we find that $f(1)=3$. This is our 'master key'. With $f(1)=3$, the function is no longer a mystery; it is fully defined as:

The Final Victory

With the function $f(n)=3n$ in our toolkit, the final steps are a victory lap. We need to find the product $f(2)\cdot f(3)$.

Using our formula:

The final product is:

Take a moment to appreciate the elegance of this journey. We started with an abstract, intimidating functional equation, and through logical deduction, we transformed it into a simple linear relationship.

This is the essence of JEE mathematics—not memorizing formulas, but understanding the 'why' behind the behavior. Keep this additive property in your arsenal; it will serve you well in many challenges to come.